Prove that the equivalence classes of an equivalence relation form a partition of the domain of...

Prove that the equivalence classes of an equivalence relation form a partition of the domain of the relation. Namely, suppose ? be an equivalence relation on a set ? and define the equivalence class of an element ?∈? to be

[?]?:={?∈?|???}.

That is [?]?=?(?). Divide your proof into the following three peices:

Prove that every partition block is nonempty and every element ? is in some block.
Prove that if [?]?∩[?]?≠∅, then ???. Conclude that the sets [?]? for ?∈? are partitions of ?.
Prove that ???⟺[?]?=[?]?.

Expert Solution

Colby Messinger answered 2 years ago

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